Characteristic analysis and motion control of a novel ball double-screw hydraulic robot joint

ABSTRACT The hydraulic joint is the key driving component of a robot. To reduce the joint size of the hydraulic robot, and improve the control accuracy and dynamic response performance, this paper proposes a novel joint structure and control method of a ball double-screw hydraulic robot. Using ball and circular arc spiral groove transmission, the hydraulic joint has a small transmission friction coefficient, compact overall structure and higher transmission accuracy. Aiming to resolve the problems of low control accuracy and motion instability caused by temperature drift in valve-controlled hydraulic systems, the high-precision joint control method based on adaptive fuzzy control compensation is used to improve the control accuracy and stability. The static and dynamic characteristics of the designed hydraulic joint are analyzed by simulation. A test platform was built, and the physical prototype of the hydraulic joint underwent static testing, dynamic control, amplitude frequency response and trajectory tracking tests. The experimental results were similar to the simulation results. The ball double-screw hydraulic robot joint has the characteristics of low starting pressure, high energy density, fast dynamic response, small amplitude frequency attenuation and high control accuracy. The starting pressure is 0.5 MPa, maximum swing frequency is 3 Hz, positioning accuracy is ± 0.03°, tracking accuracy is ± 3.9° and maximum angular velocity at 10 MPa is about 7.6 rad/s, which is close to the angular velocity of the actual human joint.


Introduction
Hydraulic joints are important parts of hydraulic manipulators, robots, military equipment and construction machinery, and are used in rehabilitation medicine and other fields. These joints are can meet the requirement of large torque, and can be used in small spaces and severe environments (Boaventura et al., 2013;Semini et al., 2011Semini et al., , 2017Shi et al., 2020;Sun et al., 2021). There are two kinds of robot hydraulic joint: the hydraulic cylinder joint and the hydraulic swing joint. Hydraulic cylindertype joints are driven by direct-acting hydraulic actuators, such as BigDog, PETMAN and the Altas series of Boston Dynamics (Griffin, 2019;Kuindersma et al., 2016;Nelson et al., 2012;Raibert et al., 2008).
Hydraulic joints are characterized by their large bearing capacity and excellent dynamic performance, but they have some shortcomings such as complex motion control and limited swing angle. The main structure of a hydraulic swing joint includes blade-type, gear-gearrack-type and double-screw-type joints. The blade-type CONTACT Meng Yang 13stek_young@tongji.edu.cn hydraulic joint has a complex structure, poor sealing performance, large internal leakage and an inability to bear high pressure, so the output torque is relatively small and the swing angle is not more than 360° (Lee et al., 2020;Xie et al., 2018). Because of the large volume of the gear-gear-rack hydraulic joint, it cannot be easily used in the mechanical arms, robots and other equipment with a limited structure size. The double-screw hydraulic joint is mainly driven by screw pair meshing, which transforms the linear motion of the piston into the rotary motion of the output shaft. It has the characteristics of compact structure, no leakage and strong compression resistance (Jiang et al., 2018). With the development of industrial technology, the hydraulic swing joint not only requires a compact structure, small size and light weight, but also has higher and higher requirements for its output torque, transmission efficiency and control accuracy (Bellicoso et al., 2018;Boaventura et al., 2012;Hyon et al., 2015Hyon et al., , 2017Liu et al., 2021). The traditional double-screw hydraulic swing joint has a large volume, the screw pair is in surface contact, the friction resistance is large, the transmission accuracy is not high and it has strong nonlinear characteristics, making is difficult to meet the requirements of high-precision control.
Compared with blade-type and gear-gear-rack-type hydraulic joints, the double-screw hydraulic joint has obvious advantages in robotic applications. Many scholars around the world have studied and optimized this type of joint. Luo et al. (2013Luo et al. ( , 2014 carried out relevant research on the double-screw hydraulic joint, proposed and optimized the joint structure, added hydraulic compensation devices and improved its nonlinear characteristics. However, many studies have not fundamentally solved the friction characteristics of the screw pair, and the friction resistance factor is still the key problem affecting its dynamic response performance and nonlinear motion control.
The screw transmission mechanism usually converts rotary motion into linear motion. The double-screw hydraulic joint is an inverse screw mechanism, which converts the linear motion of the piston into the rotary motion of the output shaft. The screw transmission mechanism can be divided into sliding friction and rolling friction transmission, according to the form of friction.
The sliding friction screw mechanism is also divided into rectangular, trapezoidal and triangular types, accor ding to the thread shape, as shown in Figure 1(a)-(c). Li et al. (2011) analyzed the flow-field characteristics of the screw pair in the hydraulic joint. According to their experiment, the factors affecting the hydraulic joint performance include the thread pitch and screwing length. The hydraulic joint studied by Li et al. is a double-screw pair with trapezoidal thread. Li and colleagues also analyzed the influence of different thread profile angles on transmission performance, and found that the trapezoidal thread with a thread profile angle of about 50°has the best response performance. When the thread profile angle increases, the transmission process becomes more stable, but when the thread profile angle is too large, the response speed becomes slower. The stress of the triangular thread is poor, which is not suitable for large load transmission. The stress of the rectangular thread is good, which is suitable for low-speed and stable transmission. The trapezoidal thread is the thread shape used in double-screw hydraulic joints at present because of its good stress and high response performance.
The rolling friction screw mechanism is mainly a ball screw structure with ball rolling, as shown in Figure 1(d). Zhou and Li (2018) analyzed the force characteristics of the ball screw, and concluded that the main factors affecting the axial force of the ball screw are the lead of the screw pair and the number of spiral grooves. Liang and colleagues (Liang, Jiang, et al., 2003;Liang, Wang, & Zheng, 2003) proposed a screw transmission mechanism, which adopts rolling screw transmission. It is found that the main factor affecting the performance of screw transmission is friction resistance, the transmission efficiency of sliding friction is low and the transmission efficiency will be significantly improved by using rolling friction transmission. The designed screw transmission mechanism is a single-screw pair, not a double-screw structure. Gong et al. (2010) analyzed the ball slip and self-locking effect in the ball screw pair and found that factors affecting the ball slip self-locking include the diameter and lead of the screw pair. The above research shows that the main factors affecting the spiral transmission of rolling friction include the friction resistance, diameter of the spiral pair, lead and number of spiral grooves.
The starting pressure of a hydraulic joint with sliding friction is large, and the starting pressure of rolling friction transmission will usually be reduced, but it has high requirements for hydraulic systems. In the valve-controlled hydraulic drive system, the electrohydraulic servo valve and other hydraulic components are affected by machining errors, oil viscosity, cleanliness and changes in environmental temperature. Therefore, there are some phenomena such as zero bias and temperature drift in the working process (Yan et al., 2019;Ma et al., 2021;Xiao, 2009). The double-screw hydraulic joint of traditional sliding friction transmission usually adopts proportional-integral-derivative (PID) control or robust control (Cao et al., 2015;Zhang et al., 2013). The zero drift and other parameters of the servo valve are not sensitive. The rolling friction transmission has high sensitivity. The drift phenomenon of the servo valve in a hydraulic system may cause poor control stability, offset and jitter during joint positioning, which will affect the accuracy of experiments on joint dynamic performance.
Taking the joint of a double-screw hydraulic robot as the research object, this paper proposes a novel joint structure of the ball double-screw hydraulic robot, which combines the double-screw structure with rolling friction. The ball and circular arc spiral groove are used for transmission, and the transmission process is rolling friction, which is similar to the ball screw structure. Compared with sliding friction, its friction coefficient is smaller, while retaining the compactness of the doublescrew structure. Aiming to resolve the problems of low control accuracy and motion instability caused by temperature drift in valve-controlled hydraulic system, the high precision joint control method based on adaptive fuzzy control compensation is used to improve the control accuracy and stability. Moreover, the static and dynamic characteristics of the designed hydraulic joint are analyzed by simulation. By building an experimental prototype and a test platform of the hydraulic joint, experiments on the starting pressure, static performance, dynamic control and tracking performance were completed. The experimental results show that the ball double-screw hydraulic robot joint has the advantages of low starting pressure, high energy density, fast dynamic response, small amplitude frequency attenuation and high control accuracy.

Structure composition
A novel type of ball double-screw hydraulic robot joint is proposed. Compared with the existing double-screw hydraulic joint, the fundamental factors that cause larger friction resistance and nonlinear motion characteristics are fully considered. The existing double-screw hydraulic joint is optimized and improved by considering the friction effect. The traditional screw pair is face contact friction. In this paper, the friction contact mode with an embedded ball is adopted. Compared with sliding friction, the rolling friction of the ball will have lower friction resistance, which may have a good impact on the nonlinear control of the hydraulic joint. The force analysis and experimental verification will be carried out later.
The prototype of the hydraulic joint is shown in Figure  2. The schematic of the ball double-screw structure for the hydraulic joint is shown in Figure 3.
In Figure 2, the hydraulic joint prototype is mainly composed of a joint servo control integrated module, elastic shaft hub, leg A and leg B. The prototype is used for static and dynamic load performance experiments  on the joints. The hydraulic joint servo control integrated module is mainly composed of the ball doublescrew hydraulic joint, electro-hydraulic servo valve, angle encoder, valve block and sealing part.
The ball double-screw hydraulic joint is the key driving component of the integrated module in Figure 2. It is mainly composed of circular arc groove screw pair, embedded ball, piston assembly, cylinder, output shaft, bearing and sealing ring. The ball double-screw structure is the main part converting the hydraulic force into the output torque, and it is also an important factor determining the conversion efficiency and control accuracy. The structural diagram of this part is shown in Figure 3.
In Figure 3, the ball double-screw structure is composed of internal and external spiral pairs. The internal screw pair is mainly composed of the output shaft, internal screw pair ball and piston assembly. The external screw pair is mainly composed of the piston assembly, external screw pair ball and cylinder. The axial position of the output shaft and the cylinder is fixed, and the piston assembly can move along the axial direction. The balls in the internal and external two-stage screw pairs are embedded in the circular arc spiral groove of the screw pair and move by means of circular rolling. The hydraulic joint pushes the piston assembly to move along the axial direction through hydraulic pressure, and the axial movement of the piston assembly is converted into the rotational movement of the output shaft through the transmission of the ball and spiral groove.
The ball double-screw hydraulic joint is driven by balls and the circular spiral groove. The ball is embedded in the spiral groove through the fixed screw (or fixed plug and retaining ring). The end face of the fixed screw (or fixed plug) has a concave spherical surface. The internal and external screw pairs are arranged in reverse, and both are multi-headed spiral grooves. A single-row ball adopts a radial arrangement, while a double-row or multi-row ball arrangement can be adopted to improve transmission stability.

Working principle
According to the rotation direction of the internal and external screw pairs, the double-screw hydraulic joint can be divided into two types: the same direction arrangement and the reverse arrangement. Considering the same swing angle, the overall volume structure of the robot joint should be as small as possible. In this paper, the internal and outer screw pairs are arranged in reverse.
The working principle and motion process of the ball double-screw hydraulic joint are as follows.
Figures 4 and 5 show the motion diagram of the hydraulic joint with the screw pairs arranged in the reverse direction. The external screw pair is left-handed and the lead is L 1 . The internal screw pair is right-handed and the lead is L 2 .
Figures 4 and 5 show the rotation of the piston assembly, cylinder and output shaft when the piston assembly moves axially in two directions. The two processes move in the opposite directions in the two figures and the principle of motion is similar. The direction shown in Figure  4 is taken as an example to analyze its working principle.  The piston moves to the right under the action of hydraulic pressure, and the movement displacement is x L . At the same time, the cylinder is fixed. The piston rotates anticlockwise under the action of the external screw pair (left-handed), and the rotation angle is θ 1 : where x L denotes the displacement of the piston to the right. Under the action of the internal screw pair (right-handed), the output shaft moves anticlockwise with respect to the piston, and the angle is θ 2 : To sum up, the output shaft rotates anticlockwise with respect to the cylinder, and the rotation angle θ is derived as follows:

Friction moment calculation
According to Equation (3), the motion relationship between the swing angular velocity ω of the output shaft relative to the cylinder block and the axial linear velocity v of the piston relative to the cylinder block can be obtained as follows: The forces of the piston and cylinder, and the piston and output shaft are shown in Figure 6, and the force analysis is described below.

Force balance analysis
In Figure 6, the force balance relationship in the vertical direction is derived as follows: where α 1 and α 2 denote the lift angle of the screw pair between the cylinder block and piston and between the piston and output shaft, respectively; A denotes the effective hydraulic area of the piston; P v denotes the driving pressure, P v = P S − P L ; and f 1 and f 2 denote the friction force of the outer screw pair and inner screw pair, respectively, f 1 = μN 1 , f 2 = μN 2 .
In Figure 6, the force balance relationship in the horizontal direction is derived as follows: According to the above equation,

Output shaft torque calculation
The tangential force acting on the output shaft by the piston is derived as follows: The output torque of the output shaft is derived as follows: where γ denotes the equivalent friction angle, μ = tan γ , and d 2 denotes the pitch diameter of the output shaft thread.
Owing to the friction resistance of the sealing ring, the actual output torque is less than in Equation (9). The hydraulic joint has three radial sealing rings, which are located at the piston and the end covers at both ends.
The maximum contact stress σ x max and the deformation contact width w x of the sealing ring are expressed as follows: Assuming that the average contact stress is half of the maximum contact stress, the resistance moment generated by the sealing ring is represented by the following expression: The actual output torque of the output shaft is proposed as follows: 3. Valve-controlled electro-hydraulic servo system

Control principle of valve-controlled electro-hydraulic servo system
In this paper, the valve-controlled electro-hydraulic servo control system is used to control the joint of the hydraulic robot. The valve-controlled electro-hydraulic servo hydraulic system is mainly composed of a quantitative pump, electro-hydraulic servo valve, overflow valve, unloading valve, filter and other hydraulic components.
The quantitative pump provides power for the system. The relief valve and the unloading valve provide safety protection. The output angle of the hydraulic robot joint is controlled by the electro-hydraulic servo valve. The control principle of the electro-hydraulic position servo control system is shown in Figure 7. The control system collects the actual rotation angle through the angle encoder, and controls the joint angle by closed-loop control. The controller transmits the calculated control signal to the servo amplifier for amplification and conversion. Then, the output angle and swing direction of the hydraulic robot joint are controlled by controlling the opening size and direction of the electro-hydraulic servo valve.
The controller adopts the adaptive fuzzy control method to realize the high-precision control of the joint. The parameters of K p , K i and K d are modified online by the fuzzy controller to improve the response time and control efficiency. The adaptive control module compensates the drift in real time to improve the joint positioning accuracy and tracking accuracy. At the same time, the error and jitter caused by temperature drift can be reduced.

Modeling and simplification of valve-controlled hydraulic joint
In Equation (13), the relationship between the hydraulic pressure and the output torque of the hydraulic robot joint is obtained. In dynamic motion control, it is necessary to consider the inertia characteristics of the electrohydraulic servo valve, hydraulic robot joints and loads, as well as the oil viscosity characteristics.
The dynamic equation of the hydraulic robot joint is described as follows: where I denotes the moment of inertia of the output shaft, b denotes the damping coefficient of the hydraulic system, k denotes the load torsional stiffness coefficient, and T f denotes the disturbance torque (load transmission friction torque, etc.). The flow continuity equation of the hydraulic robot joint is described as follows: where V t denotes the sum of the hydraulic joint volume and pipeline oil volume, β e denotes the effective elastic modulus of the hydraulic oil, C tm denotes the total leakage,ẋ L denotes the axial velocity of the piston in the hydraulic joint of the robot, and Q L denotes the valve flow. According to Equation (3), the relationship between x L and θ is expressed as follows: where K θ denotes the displacement ratio coefficient, The flow equation of the servo valve is expressed as follows: where x v denotes the displacement of the servo valve spool, K q denotes the flow gain coefficient of the servo valve, and K c denotes the flow pressure coefficient of the servo valve. Laplace transform is applied to Equations (14), (15) and (17). When M ≤ T x , θ = 0. When M > T x , the transformation results are derived as follows: where K M denotes the output shaft pressure torque coefficient, The relationship between spool displacement and control current is expressed as follows: where K v denotes the proportional coefficient of the servo valve input current. The relationship between the current and voltage of the servo amplifier is expressed as follows: where i denotes the input current of the servo valve, u denotes the control voltage of the servo amplifier, K i denotes the voltage proportional coefficient of the servo amplifier, τ i and τ u denote the time constants of Equations (19) and (20), respectively. Equations (19) and (20) can be simplified as follows: where K u = K v K i . The transfer function of the proportional part of the servo amplifier is expressed as follows: Combined with Equations (18) and (22), when M ≤ T x , θ = 0. When M > T x , the transfer function of the valve-controlled electro-hydraulic position servo system is derived as follows:

Fuzzy PID controller
The fuzzy rules corresponding to joint angle error e, error change rate e c and PID increment parameters are established by fuzzy PID to realize the real-time on-line adjustment of the control parameters.
The calculation formulas of the fuzzy PID control parameters is as follows: where K p0 , K i0 and K d0 denote the previous control parameters, and the new parameters are adjusted according to the correction value calculated by fuzzy control.
Set the range of error e and error change rate e c as [−10, 10], the range of ΔK p , ΔK i and ΔK d as [−3, 3], and the fuzzy subsets as NB, NM, NS, ZE, PS, PM, PB.

Adaptive compensation module
The moving window method is used to collect the joint motion state in recent time. The corrected compensation value after gradual convergence is obtained by calculating the minimum mean square error.
Data signals such as target angle value T(i) and actual angle value X(i) in recent time are collected through the moving window.
The minimum mean square iteration coefficient w is as follows: where u denotes the step factor, E denotes the deviation between the target angle value and the minimum mean square output value, X denotes the actual angle value, and k denotes the calculation times of each iteration in the moving window. The minimum mean square output value Y is as follows: The deviation E between the target angle and the minimum mean square output value is as follows: By cyclic iterative calculation of Equations (25)- (27), the output value Y and deviation E at the time of the moving window can be obtained. The corrected compensation value C is as follows: where C 0 denotes the compensation value of the previous correction and v denotes the correction coefficient. The system drift can be compensated adaptively through continuous correction. This method can improve the control accuracy.

Simulation
In this paper, MATLAB ® Simulink was used for simulation analysis. By establishing the simulation model, the ball double-screw hydraulic joint underwent the static test, dynamic response, amplitude frequency response and hip-knee tracking experiments of humanoid jumping motion.

Simulation parameters
According to the literature analysis and mathematical model, the main structural design parameters affecting the joint performance of a ball double-screw hydraulic robot include the lead, lift angle, screw pair diameter, friction coefficient, number of screw grooves, and so on.
These parameter values are mainly designed according to the previous experiments of this research group on the screw pair and the empirical parameters obtained by consulting the literature. On the basis of ensuring the transmission efficiency and strength of the hydraulic joint, it is also necessary to ensure the compactness of the structure as much as possible, and the lead, thread diameter and screwing length should not be too large. The basic parameters of a ball double-screw hydraulic joint are shown in Table 1. The simulation parameters of the hydraulic control system are shown in Table 2. According to the parameters and action requirements of the hydraulic joint, the AVIC FF102/30 series is selected as the electro-hydraulic servo valve, and the parameters are shown in Table 3. The screw pair adopts an embedded ball friction transmission mode, and the ball is a quenched steel ball. The friction contact area between the ball and the circular arc screw pair is small, which is rolling friction, and the clearance is adjustable. While ensuring the assembly accuracy, the preload on the spiral groove can be almost ignored and the friction force   Table 3. Parameters of the electro-hydraulic servo valve. Parameters

Reference values
Maximum current I max (mA ) 10 Maximum voltage U max (V) 10 Flow gain coefficient K q (m 3 ·s −1 /m) 0.5 Flow pressure coefficient K c (m 3 ·s −1 /Pa) 1.51 × 10 −11 Input current scale factor K v (m·A −1 ) 2 × 10 −4 Amplifier scale factor K i (mA·V −1 ) 1 Zero bias (%) ≤ ± 3 Temperature drift (%) ≤ ± 4 Frequency characteristic (Hz) ≥ 100 Working temperature (°C) −55 to +150 is small. The traditional double-screw hydraulic joint adopts trapezoidal thread transmission, and the friction contact area is large. Because of the difficulty of processing, the local pressure and friction on the spiral groove will be large while ensuring the tight assembly. In this simulation experiment, a traditional double-screw hydraulic joint, with similar volume, displacement and hydraulic action area to the designed hydraulic joint, was selected for comparison, and the performance difference between the ball rolling friction and traditional trapezoidal thread sliding friction was analyzed. The design parameters of the hydraulic joint are the values used for the experimental tests. The parameters retain some strength safety margin within the range of empirical values, which has some optimization space. The subsequent research will further optimize the structural parameters of the joint. This test only analyzes the functional realization and performance test of the double-screw hydraulic joint with a rolling friction structure.

Static simulation experiment
In the joint static simulation experiment, the input pressure of the hydraulic system was gradually increased from 0 MPa to 10 MPa, and the output torque of the hydraulic joint was observed and recorded. The input pressure when the hydraulic joint output screw started to rotate was taken as the starting pressure. The output torque of the hydraulic joint was the designed maximum output torque of the joint at an input pressure of 10 MPa. The starting pressure has a great influence on the dynamic response and accurate control of the joint. The maximum output torque has a great influence on the bearing capacity of the joint.
The static simulation test results of the ball doublescrew hydraulic joint and the traditional double-screw hydraulic joint are shown in Figure 8. Joint 1 is a ball double-screw hydraulic joint and joint 2 is a traditional double-screw hydraulic joint.
In Figure 8, the starting pressure of the ball doublescrew hydraulic joint is 0.84 MPa, and the maximum torque at 10 MPa is 65.65 Nċm. The starting pressure of the traditional double-screw hydraulic joint is 0.94 MPa, and the maximum torque is 58.41 Nċm at 10 MPa. Because the ball double-screw hydraulic joint has less friction resistance and greater output torque under the same pressure, it has less starting pressure, in theory.

Dynamic response simulation experiment
The dynamic response simulation experiment of the hydraulic joint can test the response speed of the joint. This performance parameter directly affects the response speed of the hydraulic robot joint. The system pressure was set to 10 MPa. The control signal was a step signal with amplitudes of 5-120°. The step response simulation results of the ball double-screw hydraulic joint and the traditional double-screw hydraulic joint are shown in Figure 9.
In Figure 9, compared with the traditional doublescrew hydraulic joint, the ball double-screw hydraulic joint has faster response speed, smaller overshoot and shorter stabilization time. At the step response of 0-120°, the time for the ball double-screw hydraulic joint to reach 120°for the first time is 0.39 s, and it is stable after 1.02 s. The time for the traditional double-screw hydraulic joint to reach 120°for the first time is 0.61 s, and it is stable after 1.45 s.

Amplitude frequency response simulation experiment
The simulation experiment of the amplitude frequency response of the hydraulic joint can verify whether the joint meets the motion speed and frequency requirements of a humanoid robot joint. The system pressure was set to 10 MPa. The control signal was a −60 to 60°sinusoidal signal, and the frequency was 0.1-3.5 Hz. The amplitude frequency response simulation results of the ball double-screw hydraulic joint and the traditional double-screw hydraulic joint are shown in Figure 10.
When the defined amplitude is attenuated to −3 dB (i.e. the amplitude is 0.707 times the target value), the joint cannot meet the motion requirements. In Figure 10, the amplitude frequency response of the ball  double-screw hydraulic joint is attenuated to −3 dB at 3 Hz, and the amplitude frequency response of the traditional double-screw hydraulic joint is attenuated to −3 dB at 1.5 Hz. With the decrease in friction resistance, the amplitude frequency response of the designed ball double-screw hydraulic joint is higher under the same conditions. From the above simulation results, it can be seen that compared with the traditional double-screw hydraulic joint, the ball double-screw hydraulic joint has smaller starting pressure, larger output torque under the same volume, faster dynamic response speed and smaller frequency response attenuation, making it easier to meet the motion requirements of a humanoid robot. Figure 11 shows a single hydraulic joint test prototype, which was used to test the performance of the ball double-screw hydraulic joint and to verify whether it meets the motion requirements of the humanoid robot. The single hydraulic joint test prototype is composed of a ball double-screw hydraulic joint, a hydraulic power unit, an electro-hydraulic servo valve, an angle encoder, a pressure sensor and an electronic control system. The experimental software was designed using LabVIEW software. Figure 12 shows the hydraulic joint servo control system software, which can realize the static experiment, step response experiment, sinusoidal frequency  response experiment and external input signal tracking experiment of the hydraulic joint servo control system. Figure 13 shows the process of the hydraulic joint experiment.

Static experiment
The hydraulic system pressure of the experimental platform was adjusted from 0 to 10 MPa to drive the load to rotate. At this time, the pressure is the starting pressure. The starting pressure and pressure-torque relationship of the hydraulic joint were tested and recorded, as shown in Figure 14.
The volume of the hydraulic joint has a great influence on the overall volume structure of the hydraulic robot.
The miniaturization and lightening of joints can fundamentally reduce the supporting structure, thus reducing the overall weight and the weight of the robot. Assuming the outer diameter D, length L and output torque M of the hydraulic joint, the energy density E of the hydraulic joint is defined as follows: The comparative analysis of the energy density between common hydraulic joints and the ball doublescrew hydraulic joint designed in this paper is given in Table 4.
The analysis results in Figure 14 and Table 4 can thus be obtained.  (1) The actual starting pressure of the hydraulic joint is 0.54 MPa, which is slightly lower than the simulation results. It may be that the actual friction resistance of the sealing ring is small. Compared with the traditional hydraulic joint, the starting pressure is significantly reduced.
(2) The actual output torque of the hydraulic joint is 67.95 Nċm, and the simulation result is 65.65 Nċm at 10 MPa. The actual output torque is slightly larger than the simulation results. It may be that the actual experiment has a better lubrication effect, smaller friction resistance and higher energy density. (3) The actual pressure-torque of the hydraulic joint experiment is very close to the theoretical pressure-torque obtained by simulation. The maximum absolute error occurs at 10 MPa, and the error is 2.3 Nċm. The linearity of the pressure-torque is good.

Dynamic response experiment
The system pressure was set to 5 and 10 MPa. The control signal was a step signal with amplitudes of 5-120°. The dynamic response experimental results are shown in Figures 15 and 16. It is assumed that when the amplitude error is within 0.2°, the system tends to be stable, and the experimental analysis of the dynamic response is given in Table 5. The analysis results in Figures 15 and 16 and Table 5 can thus be obtained.  (1) With the increase in pressure or amplitude, the response delay is maintained between 0.075 and 0.1 s without obvious change.
(2) With the increase in system pressure, the joint swing speed increases, the time to first achieve the target angle decreases, but the time to reach stability becomes longer. (3) With the increase in amplitude, the time to first achieve the target angle and the time to reach stability become longer. (4) At 10 MPa, the actual response delay is about 0.075-0.1 s, the time to reach 120°for the first time is 0.425 s, and the stability time of the hydraulic system  is 1.5 s, which are close to the theoretical response speeds.

Amplitude frequency response experiment
The system pressure was set to 5 and 10 MPa. The control signal was a 0-120°sinusoidal signal (i.e. amplitude 60°, amplitude offset 60°), and the frequency was 0.1-3.5 Hz. The experimental results for the amplitude frequency response of the ball double-screw hydraulic joint and the traditional double-screw hydraulic joint are shown in Figures 17 and 18 and Table 6. The analysis results in Figures 17 and 18 and Table 6 can thus be obtained.
(1) With the increase in frequency, the amplitude attenuation and angle error of the hydraulic joint increase.
(2) With the increase in system pressure, the amplitude attenuation and angle error of the hydraulic joint decrease. At 5 MPa, there is a certain delay phase difference within 1 Hz, the angle error is small, and the error becomes larger when it is greater than 1 Hz. At 10 MPa, there is a certain delay phase error within 2 Hz, the angle error is small, and the error becomes larger when it is greater than 2 Hz. (3) The attenuation of amplitude frequency response shall not be higher than −3 dB. The maximum frequency of the hydraulic joint is about 1.5 Hz at 5 MPa. The maximum frequency of the hydraulic joint is about 3 Hz at 10 MPa.
(4) The frequencies of human walking, running and jumping are about 1-2.5 Hz. When the system pressure is 10 MPa, the designed hydraulic joint can meet the joint continuous motion requirements of a humanoid robot.
The traditional double-screw hydraulic joint has high pressure resistance, large output torque and almost no leakage, but the large starting pressure seriously affects the response speed, swing frequency and control accuracy of the hydraulic joint. Moreover, this kind of hydraulic joint structure has a large volume, and is usually used for construction machinery or large hydraulic manipulators with slow rotation. The starting pressure of the blade-type hydraulic joint is small, and the swing frequency is higher than 3 Hz. It can meet the movement speed requirements of a humanoid robot, but it is also large, with complex sealing, can be difficult to be apply in high-pressure environments and the capacity density is low.
Compared with the traditional double-screw hydrau lic joint, the ball double-screw hydraulic joint designed in this paper has less friction resistance and lower starting pressure. It can realize a swing frequency similar to that of the blade swing joint, while maintaining the advantages of a compact structure and large output torque.

Control accuracy experiment
According to previous experiments, the minimum control current of a servo valve required for hydraulic joint start-up is 0.1 mA under 10 MPa pressure. The model of servo valve used in the hydraulic system is the AVIC FF-102/30 series. According to the parameters of the servo valve in Table 3, the maximum current change caused by temperature drift is ± 0.4 mA within the working temperature range of −55 to +150°C. The current change caused by temperature drift exceeds the minimum current required for hydraulic joint control, which will have a great impact on the control of the hydraulic joint. In this paper, adaptive fuzzy control is used to compensate for the temperature drift to improve the control accuracy.     Table 7 can thus be obtained.
(1) Under different temperature environments, the positioning accuracy of the joint can be controlled within ± 0.03°and the tracking accuracy can be controlled within ± 3.9°. (2) The joint structure and adaptive fuzzy control method have high dynamic control accuracy.

Knee and hip joint tracking experiment of robot humanoid jumping motion
The hydraulic joint needs to meet the continuous motion requirements of a humanoid robot, and also needs to meet the explosive speed requirements (Chen et al., 2021;Chevallereau et al., 2021;He & Geng, 2011;Ishihara et al., 2020;Miyadaira et al., 2018;Vukobratovi et al., 2007). In common walking, running and jumping, jumping requires the highest short-term explosive speed of joints (Ahn & Cho, 2018Tajima & Suga, 2006; Xu et al., 2008, 2012). Owing to the function of the accumulator in the hydraulic system, the hydraulic joint can provide short-term discontinuous or intermittent rapid rotation, which will exceed the rotation speed in the amplitude frequency response (Zhang et al., 2017, pp. 2-17). The motion of human jumping motion is selected as the trajectory target of the following experiment, which can verify the dynamic performance of the designed joint in humanoid jumping motion.
A wireless IMU attitude sensor was used to capture human jumping motion. The wireless IMU sensor used an ALUBI LPMS-B2 module to obtain the motion angle and angular velocity data of the knee and hip joints during human jumping through multi-sensor synchronous measurement. The data for jumping target tracking are from the author's published papers (see Bian et al., 2021).
The hip and knee tracking experiments of the hydraulic joint jumping motion are shown in Figures  21 and 22. An analysis of the experimental results is presented in Table 8.
The analysis results in Figures 21 and 22 and Table 8 can thus be obtained.   (1) The angle tracking errors of the knee and hip joints are small. When the system pressure is low, the joint rotation is smoother, and the angle tracking error is smaller in the squatting and take-off stages. At 5 MPa, the knee joint has the maximum angle error of −0.024°in the squatting stage and −2.992°in the take-off stage, and the hip joint has the maximum angle error of −0.526°in the squatting stage and 0.045°in the take-off stage.
(2) The angular velocity errors of the knee and hip joints are small. When the system pressure is high, the hydraulic joint has a large maximum angular velocity and small angular velocity error. At 10 MPa, the knee joint has a maximum angular velocity error of 0.064 rad/s in the squatting stage and −0.032 rad/s in the take-off stage, and the hip joint has a maximum angular velocity error of 0.235 rad/s in the squatting stage and −1.222 rad/s in the take-off stage. (3) The designed hydraulic joint has a high explosive speed. When the system pressure is 10 MPa, the maximum angular velocity in the take-off stage is 6.639 rad/s for the knee joint and 7.635 rad/s for the hip joint.
The designed ball double-screw hydraulic joint has good performance in tracking hip and knee joints. It can  Table 9. Characteristic parameters of the ball double-screw hydraulic robot joint.

Characteristic parameters Values
Starting pressure (MPa) 0.54 Maximum output torque at 10 MPa (N·m) 67.95 Response delay (ms) ≤ 100 Frequency response at −3 dB (Hz) 3 Positioning accuracy (°) ± 0.03 Tracking accuracy (°) ± 3.9 Maximum angular velocity at 10 MPa (rad/s) 7.6 provide faster rotation speed when the pressure is high, and its maximum speed can reach the maximum speed of hip and knee jumping. According to the above experiments, the performance parameters of ball double-screw hydraulic joint are shown in Table 9.
The characteristic parameters of the hydraulic joints in Table 9 will form the basis for future research on joint control technology and hydraulic robot design.

Conclusion
Aiming to meet the motion performance requirements of hydraulic robot leg joints, this paper proposes a novel joint structure and control method of a ball double-screw hydraulic robot. The following conclusions are obtained through simulation and prototype experiments.
(1) The friction resistance and structure volume of the double-screw hydraulic joint are reduced by the ball and arc spiral groove transmission. This structure increases the output torque of the joint. The maximum torque of 10 MPa is 67.96 Nċm, and the energy density is 0.199 Nċm/m 3 . (2) By building the experimental prototype and test platform of the hydraulic joint, experiments of the starting pressure test, static performance, dynamic characteristics and tracking performance were completed. The designed hydraulic joint has good control accuracy and response frequency. The starting pressure is 0.5 MPa, the maximum swing frequency is 3 Hz, the positioning accuracy is ± 0.03°and the tracking accuracy is ± 3.9°.
(3) Using motion data from human joints, motion tracking experiments of the hip and knee joints during jumping were completed. From the experimental results, the hydraulic joint has small tracking error and higher swing angular velocity, and the maximum angular velocity at 10 MPa is about 7.6 rad/s, which is close to the angular velocity of the actual human joint.
The hydraulic joint designed in this paper mainly focuses on the novel structural design and the realization of control function. The above results and experiments provide a basis for further research on joint control algorithms and hydraulic robot design.
Although the ball double-screw hydraulic joint and the control method proposed in this paper are designed for the joints of hydraulic robots, the good dynamic performance and compact structure of this joint make it suitable for other situations with high requirements for joint performance and compact size.